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Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula

Author

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  • Xavier Bardina

    (Universitat Autònoma de Barcelona)

  • Maria Jolis

    (Universitat Autònoma de Barcelona)

Abstract

We prove a uniform bound for the density, p t (x), of the solution at time t∈(0, 1] of a 1-dimensional stochastic differential equation, under hypoellipticity conditions. A similar bound is obtained for an expression involving the distributional derivative (with respect to x) of p t (x). These results are applied to extend the Itô formula to the composition of a function (satisfying slight regularity conditions) with a hypoelliptic diffusion process in the spirit of the work of Föllmer et al. (5)

Suggested Citation

  • Xavier Bardina & Maria Jolis, 2002. "Estimation of the Density of Hypoelliptic Diffusion Processes with Application to an Extended Itô's Formula," Journal of Theoretical Probability, Springer, vol. 15(1), pages 223-247, January.
    • Handle: RePEc:spr:jotpro:v:15:y:2002:i:1:d:10.1023_a:1013899603656
    DOI: 10.1023/A:1013899603656
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    References listed on IDEAS

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    1. Rozkosz, Andrzej, 1996. "Stochastic representation of diffusions corresponding to divergence form operators," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 11-33, October.
    2. María Emilia Caballero & Begoña Fernández & David Nualart, 1998. "Estimation of Densities and Applications," Journal of Theoretical Probability, Springer, vol. 11(3), pages 831-851, July.
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